Heterozygote advantage as a natural consequence of
adaptation in diploids
Diamantis Sellisa, Benjamin J. Callahanb, Dmitri A. Petrova,1, and Philipp W. Messera,1
Departments of aBiology and bApplied Physics, Stanford University, Stanford, CA 94305
Edited* by Boris I. Shraiman, University of California, Santa Barbara, CA, and approved November 2, 2011 (received for review September 7, 2011)
Molecular adaptation is typically assumed to proceed by sequential fixation of beneficial mutations. In diploids, this picture
presupposes that for most adaptive mutations, the homozygotes
have a higher fitness than the heterozygotes. Here, we show that
contrary to this expectation, a substantial proportion of adaptive
mutations should display heterozygote advantage. This feature of
adaptation in diploids emerges naturally from the primary importance of the fitness of heterozygotes for the invasion of new
adaptive mutations. We formalize this result in the framework of
Fisher’s influential geometric model of adaptation. We find that in
diploids, adaptation should often proceed through a succession of
short-lived balanced states that maintain substantially higher levels of phenotypic and fitness variation in the population compared
with classic adaptive walks. In fast-changing environments, this
variation produces a diversity advantage that allows diploids to
remain better adapted compared with haploids despite the disadvantage associated with the presence of unfit homozygotes. The
short-lived balanced states arising during adaptive walks should
be mostly invisible to current scans for long-term balancing selection. Instead, they should leave signatures of incomplete selective
sweeps, which do appear to be common in many species. Our
results also raise the possibility that balancing selection, as a natural consequence of frequent adaptation, might play a more prominent role among the forces maintaining genetic variation than is
commonly recognized.
A
daptation by natural selection is the key process responsible
for the fit between organisms and their environments. The
invasion of new adaptive mutations is an essential component of
this process that fundamentally differs between haploid and
diploid populations. In diploids, while a new mutation is still
rare, natural selection acts primarily on the mutant heterozygote
(1). As a result, only those adaptive mutations that confer a fitness advantage as heterozygotes have an appreciable chance of
invading the population (“Haldane’s sieve”). However, if the
heterozygote of an invading adaptive mutation is fitter than the
mutant homozygote, the mutation will not be driven to fixation
but, instead, maintained at an intermediate, balanced frequency.
We argue that heterozygote advantage should be very common
during adaptation in diploids if selection is stabilizing and at least
some mutations are large enough to overshoot the optimum.
Consider, for example, adaptation through changes in gene expression (Fig. 1A), which is an important, if not the dominant,
mechanism of adaptation (2). Here, mutations of small effect
(Fig. 1B) will be adaptive when they modify expression in the
adaptive direction whether the organism is haploid or diploid.
However, mutations of large effect (Fig. 1C) can be nonadaptive
in haploids, because they overshoot the optimum, yet be adaptive
in diploids, because their phenotypic effect is moderated when
heterozygous. These mutations will have heterozygote advantage
and lead to balancing selection in diploids.
The components of the simple model in Fig. 1, stabilizing selection and availability of mutations of various sizes, are well
established by empirical data in the case of gene expression.
Pervasive stabilizing selection is indicated by the lack of large gene
expression differences between and within species despite the
abundance of mutations that change gene expression (3–7). In
20666–20671 | PNAS | December 20, 2011 | vol. 108 | no. 51
addition, expression-altering mutations are known to come in a
variety of sizes ranging from subtle changes to dramatic ones of
tens of fold or even hundreds of fold (4, 5, 8).
The model in Fig. 1, albeit instructive, may lack generality because it incorporates only a single trait. However, adaptation
might often involve mutations that have complex pleiotropic
effects in an effectively multidimensional phenotypic space. The
classic model that incorporates this key feature of adaptation is
Fisher’s geometric model (9–13). In this single-locus model,
phenotypes are vectors in an abstract geometric space (Fig. 2A),
with the orthogonal axes representing different, independent
traits, such as color and height. Mutations move phenotypes in
a random direction, with the size of the step corresponding to the
effect size of the mutation. The fitness landscape is peaked around
a single optimal phenotype, capturing our intuition of stabilizing
selection. Below, we extend Fisher’s geometric model to diploidy
and confirm the intuition from Fig. 1 that adaptation should generically lead to heterozygote advantage.
Results
Consider an allele a with homozygous phenotype raa in Fisher’s
model. Mutations are modeled by adding a mutation vector m to
the phenotype of the mutated allele (Fig. 2A). The direction of
the mutation vector is chosen uniformly, and its size (m) is drawn
from a specified distribution P(m). In diploids, the phenotype is
determined by its two constituent alleles. The geometric nature
of Fisher’s model offers a straightforward mapping between the
phenotype of the mutant heterozygote (rab) and the homozygote
phenotypes (raa and rbb = raa + m) in terms of the weighted
average: rab = raa + hm, where the weight h specifies the phenotypic dominance of the new mutation. For convenience, we
assign the phenotype of a diploid homozygote to be equal to that
of a haploid carrying the same allele (i.e., rii = ri).
Fitness w(r) decreases with the distance between the organismal phenotype and the phenotypic optimum. In haploids, adaptive mutations, w(rb) > w(ra), are those that bring the mutant
allele closer to the optimum (i.e., the mutant falls inside the
sphere αhap of radius ra centered at the optimum) (Fig. 2A). In
diploids, it is the fitness of the mutant heterozygote rab that primarily determines the probability of successful invasion; therefore, we define the area in which mutations are adaptive in
diploids (αdip) by the condition w(rab) > w(raa).
Let us first focus on the instructive case of strict phenotypic
codominance, h = 1/2. Here, mutations in diploids have half
the initial phenotypic effect that they would have in a haploid.
Author contributions: D.A.P. and P.W.M. designed research; D.S., B.J.C., and P.W.M.
performed research; D.S., B.J.C., and P.W.M. contributed new reagents/analytic tools;
D.S., B.J.C., D.A.P., and P.W.M. analyzed data; and D.S., B.J.C., D.A.P., and P.W.M. wrote
the paper.
The authors declare no conflict of interest.
*This Direct Submission article had a prearranged editor.
Freely available online through the PNAS open access option.
1
To whom correspondence may be addressed. E-mail: dpetrov@stanford.edu or messer@
stanford.edu.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
1073/pnas.1114573108/-/DCSupplemental.
www.pnas.org/cgi/doi/10.1073/pnas.1114573108
A
B
C
fitness
small mutation (1.5-fold change)
large mutation (3-fold change)
new
old
mut (hap)
1.5
mut (hap)
3
3
1
2
expression level
0.75
wt (hap)
1
het (dip)
0.5
wt (dip)
0.5
0.5
1.5
het (dip)
0.75
hom (dip)
0.75
0.5
1.5
hom (dip)
1.5
Mutations of small effect thus have reduced initial selective effects
in diploids compared with in haploids, but the area in which
mutations are adaptive in diploids (αdip) is significantly enlarged
(Fig. 2A). If the supply of mutations is restricted to those of small
size compared with the distance to the optimum, adaptive mutations will invade a diploid population at a slower rate than they
would a haploid population because of the reduced initial selection (SI Text and Figs. S1 and S2). In contrast, once mutations of
large enough size are available, diploids start reaping the benefits
of the larger space of adaptive mutants available to them. In this
case, adaptive mutations will invade at a higher rate in diploids
than in haploids, even after controlling for the difference in mutation rate that results from a diploid population having twice as
many chromosomes as a haploid population of equal size (14).
The larger range of adaptive mutations available to diploids
comes with a catch, however; many of these adaptive mutations
display heterozygote advantage, and thus will not simply go to
fixation. The fraction δu of adaptive mutations with heterozygote
advantage [w(raa) < w(rab) > w(rbb)] in Fisher’s model is the
fraction of mutations that fall into the sphere αdip [w(rab) >
w(raa)] but not the sphere γ [w(rbb) > w(rab) > w(raa)] (Fig. 2A).
In the small-mutation limit, δu goes to zero because the nonoverlapping space between αdip and γ is not reachable by mutations that fall in the immediate neighborhood of ra. In the limit
in which mutations of all sizes are equally likely, however, most
adaptive mutations show heterozygote advantage (δu ≥ 1/2),
except for the special cases of perfect dominance or perfect recessiveness (SI Text).
The crucial criterion determining the probability of heterozygote advantage during adaptation in diploids is the availability of
“large” mutations. In Fisher’s model, for heterozygote advantage
to be common, the average size of mutations (<m>) has to be at
least of order raa/√d, the distance to the optimum divided by the
square root of the number of phenotypic dimensions (SI Text).
However, even when mutations are initially small compared with
this distance, during an adaptive walk, a population gradually
approaches a fitness optimum via successive adaptations (11). Thus,
at some point on the walk, this condition will be met; thereafter,
heterozygote advantage will be likely.
We performed simulations of adaptive walks in Fisher’s model
to test these theoretical predictions and further investigate the
consequences of frequent heterozygote advantage in such walks.
Specifically, we simulated a single locus in a 2D phenotypic space
Sellis et al.
under phenotypic codominance in a Wright–Fisher framework
using an exponential distribution of mutation sizes and a symmetrical Gaussian fitness landscape (Materials and Methods).
We find that adaptive walks in diploids typically involve the
succession of many intermediate balanced polymorphisms that
tend to be ephemeral; they are quickly displaced by new adaptive
alleles, themselves often displaying heterozygote advantage (Fig.
2 B and C). These dynamics contrast sharply to those in haploid
populations, where adaptive walks proceed by successive sweeps
of adaptive mutations and populations are generally monomorphic between sweeps (Fig. 2C).
We confirm that balanced states are likely during adaptive
walks (Fig. S3), provided that mutation sizes meet our theoretical conditions and selection is strong enough to maintain balanced states despite the stochastic fluctuations arising from genetic
drift (SI Text). Fig. 2D shows the probability of observing balanced states during adaptive walks under various parameter
settings (Table S1). Note that the probability of observing balanced states correlates very strongly with the overall probability
of observing successful adaptation. For example, in all our scenarios where the population eventually traversed at least 90% of
the initial fitness distance to the optimum, balanced states were
observed in ≥90% of the runs and were present ≥40% of the
time during walks.
Simulated haploid populations approach the optimum faster,
on average, than simulated diploid populations despite diploid
populations generally containing the most fit individual (Fig. S4).
This is because diploid populations suffer from a high genetic
load, which can be identified as the segregation load attributable
to pervasive heterozygote advantage. The balanced polymorphisms that result from the invasion of adaptive mutations with
heterozygote advantage also cause high levels of standing variance in both phenotype and fitness to persist during the adaptive
walk (Fig. S4).
The maintenance of genetic variation via heterozygote advantage during adaptation is a striking difference between the
adaptive walks in haploids and diploids. Although the load
resulting from the maintained variation is costly during adaptation to a constant environment, could this effect be advantageous
in a changing environment? Environmental changes that displace the phenotypic optimum can convert dominance variance
of fitness into additive variance, which would then fuel adaptation. To investigate this possibility, we incorporated a randomly
PNAS | December 20, 2011 | vol. 108 | no. 51 | 20667
EVOLUTION
Fig. 1. Adaptation to a change in the optimal level of gene expression. (A) In both haploids (hap) and diploids (dip), the wild type (wt) is perfectly adapted to
the original fitness function (dashed black curve). After an external change, the optimal expression level becomes twice the original level (solid red curve).
Note that we are assuming phenotypic codominance; thus, the two individual gene copies in a diploid each contribute expression level 0.5, such that overall
expression is 1. (B) Fitness effects of a small mutation that increases expression level by 1.5-fold. The mutant heterozygote (het) is less fit than both the
haploid mutant (mut) and the mutant homozygote (hom). (C) Effects of a large mutation that increases expression by threefold. In this case, the mutant
heterozygote effectively has only twofold increased expression, and thus lands right at the new fitness optimum. In contrast, both the haploid mutant and
the mutant homozygote “overshoot” the optimum.
A
B
C
D
Fig. 2. Fisher’s geometric model of adaptation in two dimensions. (A) Two orthogonal axes represent independent character traits. Fitness is determined by
a symmetrical Gaussian function centered at the origin. Consider a population initially monomorphic for the wild-type allele raa = (2,0). A mutation m gives
rise to a mutant phenotype vector rbb = raa + m. The phenotype of the mutant heterozygote assuming phenotypic codominance (h = 1/2) is rab = raa + m/2. The
different circles specify the areas in which mutations are adaptive in haploids (αhap), adaptive in diploids (αdip), and replacing in diploids (γ). (B) Frequency
trajectories of all alleles present during a representative adaptive walk in a diploid population with N = 5·104, raa = (2,0), and <m> = σ w = 1. Different colors
represent different alleles. The black bars over the graph indicate the periods during which a balanced polymorphism was present. (C) Representative
adaptive walks in a haploid population and a diploid population. Vectors depict the successive mutations that led to the prevalent allele at the end of the
walk. The haploid walk consists of a single lineage of successive mutations, each conferring a selective advantage over the previous one. In the diploid walk,
the first mutation overshoots the fitness optimum, generating a sequence of intermediate balanced states. Note that the areas αhap, αdip, and γ (dotted circles)
from A apply only to the first mutation in the walk when the population is still monomorphic for ra. (D) Probability of observing balanced polymorphism
during adaptive walks toward a fixed fitness optimum as a function of mutation sizes scaled by effective drift radius r0 (SI Text) for the various settings of N,
σ w, and <m> specified in Table S1. Circles show the probability of at least one balanced state arising over the course of a walk, and squares show the fraction
of time during which balanced states were present. Coloration indicates the average “adaptedness” achieved during a walk, defined by the improvement in
mean population fitness over the walk (<wend> − <wstart>) relative to the maximally possible improvement (1 − <wstart>).
moving optimum into our simulations (Materials and Methods).
The optimum moves every generation in a random direction with
step sizes sampled from a Gaussian distribution, the variance of
which (σ env2) determines the speed of optimum movement.
Pervasive balanced polymorphism remains a feature of adaptation in diploid populations in the moving optimum scenario
(typical trajectories are shown in Fig. S5). In simulations with a
fast-moving optimum (σ env/σ w = 10−2), polymorphisms at frequencies 0.05 < x < 0.95 are present around 30% of the time
when averaged over replicate runs. More than 80% of these
polymorphisms are balanced. Polymorphisms are observed less
frequently (∼5%) when environmental change becomes very
slow (σ env/σ w = 10−5), because populations are well adapted
most of the time. The fraction of polymorphisms that are balanced, however, remains substantial (∼58%) in this scenario.
Table S2 shows the percentages of time during which polymorphisms were observed in our simulations and the fractions of
those polymorphisms that were balanced for a wide range of
values of σ env/σ w. We also demonstrate that many of the balanced polymorphisms eventually fix such that a substantial
20668 | www.pnas.org/cgi/doi/10.1073/pnas.1114573108
proportion of substitutions (60–70%) pass through a balanced
state (Fig. S6 and Table S2). These substitutions tend to be
generated by larger phenotypic effect mutations than the substitutions that go to fixation without an intermediate balanced
state (Fig. S6).
To evaluate how effectively populations followed the moving
fitness optimum, we measured the mean population fitness averaged over a walk (<w>). The difference, λ = 1 − <w>, is then
the average lag in fitness between the population and the optimum. Fig. 3A shows the ratio λhap/λdip between haploid and
diploid populations for different values of σ env/σ w. In slowchanging environments (σ env/σ w < 10−4), haploids follow the
moving optimum more closely than diploids, replicating the
constant environment result. However, in fast-changing environments (σ env/σ w > 10−4), it is diploids that prevail.
The diversity advantage of diploid populations derives from
the greater variation maintained during adaptation. Specifically,
this greater variation should lead to an increased range of
mutations starting from multiple balanced alleles as well as the
ability of diploids to adapt by fast adjustment of the frequencies
Sellis et al.
B
C
Fig. 3. Statistics of adaptive walks under a moving fitness optimum. (A)
Ratio of the average lag in fitness (λ) between the population and the optimum in haploids (hap) and diploids (dip) as a function of the speed of
environmental change. In fast-changing environments, diploid populations
follow the moving optimum more closely than haploids (λhap/λdip > 1). (B)
Fitness variance attributable to balanced polymorphisms (frequency 0.05 <
x < 0.95) and the age of the balanced polymorphism for different values of
σ env. Both quantities are estimated from the balanced polymorphisms that
were present at the end of simulation runs. The age of a balanced polymorphism is defined as the time since the most recent common ancestor of
its constituent alleles. Data points are medians over 103 runs, and error bars
specify the 10% and 90% quantiles. The gray-shaded area (0.04N < age <
4N) indicates the expected age range of common neutral polymorphisms at
frequencies between 0.05 < x < 0.95. (C) Same as in B but phenotypic variance is shown instead of fitness variance.
of balanced alleles without waiting for de novo mutations. Note
that this diversity advantage of diploids is different in kind from
the advantages based on the differences in number of chromosomes and the efficacy of selection in haploids and diploids
discussed previously (14–16).
The characteristics of the genetic variation in diploid populations depend strongly on the speed of environmental change.
Fig. 3 B and C show the joint distributions of allele age and the
fitness or phenotypic variance of balanced polymorphisms at
intermediate population frequencies as a function of σ env/σ w. In
slow-changing environments, balanced alleles generally persist
over long periods of time, and thus tend to be much older than
neutral alleles of comparable population frequencies. These old
balanced alleles are similar phenotypically, and their segregation
generates only limited variation in fitness (Fig. 3B) and
Sellis et al.
phenotype (Fig. 3C). In contrast, in fast-changing environments,
balanced alleles are much younger than comparably frequent
neutral alleles. These young balanced alleles are often very distinct phenotypically, and thus produce high levels of standing
variation in fitness (Fig. 3B) and phenotype (Fig. 3C).
Discussion
Our finding that heterozygote advantage emerges naturally in
Fisher’s geometric model if mutations are sufficiently large is
very robust to the details of the model, such as the number of
dimensions; the choice of phenotypic dominance rules, including
under-, over-, and incomplete dominance; mutation rate and size
distribution; population size; and flatness of the fitness function
(SI Text, Figs. S1–S3, S7, and Table S1).
The features that underlie pervasive heterozygote advantage in
Fisher’s model are also likely to apply very generally in nature: (i)
mutations in diploids should initially segregate as heterozygotes,
(ii) selection should be stabilizing for some traits, and (iii) some
invading mutations should be large enough to overshoot the local
optimum. Indeed, rare mutations are generally heterozygous,
except for cases of exceptionally strong inbreeding. It is also well
established that stabilizing selection acts on many phenotypic
traits (17, 18). Even if “more is always better” holds for certain
traits, as long as adaptive mutations generally influence at least
one trait under stabilizing selection, we still expect heterozygote
advantage to be frequent. Finally, mutations of large phenotypic
effect have been observed in many organisms, and it is becoming
increasingly apparent that such large mutations do contribute to
adaptation. For example, adaptive mutations of large effect have
occurred in the domestication of maize (19) and dogs (20), the
evolution of sticklebacks to fresh water habitats (21, 22), pesticide
resistance in insects (8), coat color in mammals (23, 24), and
several studies of experimental evolution (25, 26). Furthermore,
analyses of mutation accumulation lines have demonstrated the
availability of mutations causing multiple-fold changes in gene
expression (4, 5). Because the expression of most genes is known
to be constrained by stabilizing selection over much narrower
ranges (3–7), we must conclude that gene expression mutations
that overshoot the optimum are likely abundant.
The classic model of adaptation holds that adaption is driven
by adaptive mutations that sweep quickly to fixation. Although
our model also predicts such fast fixation events, it additionally
predicts that many adaptive mutations will initially only sweep to
intermediate frequencies, where they are then maintained for
a period of time by balancing selection, before continuing on to
either fixation or loss. Both models thus predict an elevated rate
of fixation at functional sites compared with the neutral expectation and a local reduction of genetic diversity around adaptive
sites, as have been observed in a range of organisms (27–30).
However, in contrast to the classic model, we also expect the
presence of many incomplete selective sweeps. Indeed, the genomic signatures of such incomplete sweeps appear to be plentiful in a number of organisms (31–33). Incomplete sweeps also
appear to be common in experimental evolution in Drosophila,
where virtually no classic sweeps have been detected after 600
generations of evolution despite abundant evidence of phenotypic adaptation over this period (34).
The abundance of incomplete sweeps in natural and experimental populations is consistent with but, unfortunately, not
uniquely predictive of adaptation-driven balancing selection.
Other scenarios, such as frequency-dependent selection, adaptation to specific subhabitats, and polygenic adaptation, also predict
incomplete sweeps. The only way to test the hypothesis of heterozygote advantage explicitly is to measure fitness of the homozygotes and heterozygotes for the putatively balanced alleles
directly. Such measurements are difficult but not impossible and
can now be carried out systematically in laboratory systems of
artificial selection, such as yeast or Drosophila (26, 34, 35).
PNAS | December 20, 2011 | vol. 108 | no. 51 | 20669
EVOLUTION
A
However, in many other systems, direct fitness measurements
are not feasible and balanced polymorphisms have to be identified
by alternative approaches. The standard scans for balanced polymorphisms are inappropriate for our purposes because they typically search for very ancient balanced alleles (36–38), whereas the
balanced alleles predicted by our model are often short-lived. To
identify young balanced alleles specifically, one can search for
polymorphisms that maintain their frequencies in the face of sharp
bottlenecks (39). A particularly powerful system in this context is
provided by human genetics because of the multiple recent bottlenecks associated with human migrations. Other opportune
systems are island species recovering from natural disasters or
man-made relocations of species to new but environmentally
similar locations, such as the relocation of Euphydryas gillettii from
Wyoming to Colorado (40).
Adaptive evolution is generally thought to be antithetical to
the maintenance of genetic variation. In contrast, in our model,
pervasive adaptation systematically generates genetic variation
by promoting balanced polymorphisms. These balanced polymorphisms are expected to segregate at high frequencies yet can
affect both phenotype and fitness substantially. This is very different from the common view that frequent polymorphisms
should be neutral or subject to only weak selective forces.
We argue that adaptation-driven balanced polymorphisms can
be an important source of consequential genetic variation. In
particular, we believe that the balanced polymorphisms predicted
by our model can be associated with human disease. Some of the
common disease variants could be mutations that are maintained
at high population frequencies because of strong heterozygote
advantage, although they are very harmful as homozygotes.
Balancing selection and, specifically, the prevalence of heterozygous advantage was once considered the dominant force maintaining variation in natural populations (41–43). This view fell out
of favor with the rise of the neutral theory in the 1960s (10). The
neutral theory postulates that only a small proportion of substitutions are adaptive and that these substitutions are fixed very
quickly; this, in turn, implies that practically all polymorphisms
should be either neutral or slightly deleterious. However, recent
genomic evidence has suggested that the rate of adaptation is
substantial in some organisms, with, for example, ∼50% of all
amino acid substitutions in Drosophila driven by positive selection
(28). Here, we argue that such a high rate of adaptation in diploids
should also lead to a high rate at which balanced polymorphisms
are driven into the population. We thus advocate that the balance
theory of genetic variation should be given new life and reassessed
using all the modern genomic tools at our disposal.
Materials and Methods
Monte Carlo Simulations of a Single Adaptive Mutation in Fisher’s Model. We
investigate the evolution of a single locus in Fisher’s geometric model (9).
Alleles are represented by vectors (r) in an abstract, d-dimensional Euclidean
phenotype space. Mutant alleles are obtained by adding a mutation vector
(m) to the parental allele: rb = ra + m. The directions of mutation vectors are
distributed uniformly; mutation sizes (m) are distributed according to a probability distribution P(m) with an average mutation size <m>. We consider two
such distributions in detail, the uniform distribution P(m) ∝ uniform(0,2<m>)
and the exponential distribution P(m) ∝ exp(−m/<m>). Haploid organismal
phenotypes are equal to the allelic phenotype. In diploids, the phenotype is a
weighted average of its two constituent alleles. In the case of the heterozygous mutant, the phenotype can be expressed as rab = ra + hm, with h being
the phenotypic dominance of the mutant allele.
The fitness of a phenotype is determined by its distance from the fitness
optimum r*. Following precedent (10, 11), we use a Gaussian fitness function:
w(r) = exp[−(r-r*)2/(2σ w2)]. For convenience, we set the origin of the phenotype space to be at r* and choose the scale of the space such that σ w2 = 1. We
begin by considering mutations arising in a population that is monomorphic
for the wild-type allele ra = (2,0,. . .,0). The invasion probability of new
mutations is then approximately π hap(m) = 2[w(rb)/w(ra) − 1] in haploids (10).
In diploids, we assume that mutants initially exist only as heterozygotes,
therefore: π dip(m) = 2[w(rab)/w(ra) − 1].
20670 | www.pnas.org/cgi/doi/10.1073/pnas.1114573108
Results in Figs. S1 and S2 were obtained from Monte Carlo simulations of
the above model based on 107 randomly drawn mutations per data point.
The ratios udip/uhap of the rates at which adaptive mutations occur in diploids vs. haploids were estimated by counting the overall numbers of
mutations where rb ∈ αhap in haploids or, respectively, rab ∈ αdip in diploids.
For the ratios vdip/vhap of the rates at which adaptive mutations invade the
population, each adaptive mutation was additionally weighted by its respective invasion probability, π hap or π dip. Among successfully invading
mutations, the expectation values, <w(rab) − w(ra)> in diploids and <w(rb) −
w(ra)> in haploids, were measured to estimate the ratios <Δwdip>/<Δwhap>.
Estimates of δu were obtained by counting the fraction of adaptive mutations with heterozygote advantage in the diploid scenario. For δv, each
adaptive mutation was thereby additionally weighted by its respective
invasion probability.
Simulation of Adaptive Walks Toward a Fixed Fitness Optimum. To investigate
adaptive walks toward a fixed fitness optimum, we simulated the full stochastic population dynamics in the above scenario under an infinite alleles assumption. We focused on the instructive case of a 2D Fisher’s model with complete
phenotypic codominance (h = 1/2). The phenotype of a heterozygous diploid is
then always the coordinate-wise average of its two alleles: rab = (ra + rb)/2.
Mutations are modeled by a Poisson process with rate μ = 2.5·10−7 per
individual and generation. Mutation directions are drawn uniformly, and
mutation sizes are sampled from an exponential distribution with mean
<m> = 1. Population sizes are Nhap = 105 for haploids and Ndip = 5·104 for
diploids, ensuring that new mutations arise at equal overall rates in the two
populations (Θ = 2cNμ = 0.05, where c is ploidy).
The state of the population at any given time point is specified by the set of
alleles {ri} present in the population and their associated population frequencies {xi}. Allele frequency dynamics are modeled in a Wright–Fisher
framework with selection (44). For haploids, we use the standard Wright–
Fisher sampling procedure in which allele frequencies xi(t + 1) in the next
generation are drawn from a multinomial distribution P(N,{pihap(t)}) with
selection-adjusted probabilities: pihap(t) ∝ w(ri)xi(t). In the case of diploids, we
first convert allele frequencies into genotype frequencies (assuming Hardy–
Weinberg equilibrium) to calculate the selection-adjusted probabilities:
pidip(t) ∝ ∑jw(rij)xi(t)xj(t). Allele frequencies xi(t + 1) are then drawn from
P(2N,{pidip(t)}). In both cases, pi and xi are normalized such that ∑ipi = ∑ixi = 1.
As specified above, simulations start from a population that is monomorphic for the wild type ra = (2,0) with the optimal phenotype located at
the origin, yielding an initial population average fitness of w(ra) ∼0.13.
Populations are then evolved for 104 generations, which typically suffices to
approach the fitness optimum closely (<w> > 0.96 at end of a run; Fig. S4A).
Simulations Under a Moving Fitness Optimum. For the analysis of the moving
optimum scenario, we adjust our simulation as follows. At the start of the
simulation, the population is initialized to be monomorphic for the optimal
phenotype: r*(t = 0) = ra = (0,0). In each subsequent generation, the optimum
r*(t) moves one step in a random direction and the size of the mutation is
sampled from the positive half of a Gaussian distribution with variance σ env2. In
a single simulation run, the population is evolved for 107 generations (∼100N).
We exclude the first 105 generations of each run from our analysis as a “burnin” period so as to remove the influence of the initial state of the population.
Ascertainment of Balanced Polymorphisms During Adaptive Walks. Balanced
polymorphisms can consist of several alleles (45, 46). We determine the presence of a balanced polymorphism at a given time point in our simulation runs
using Kimura’s analytic conditions (47). Assume that n alleles {r1,. . .,rn} are
present in the population with diploid fitness values given by w(rij). Let T be the
matrix defined by Tij = w(rij) − w(rin) − w(rjn) + w(rnn) (i,j = 1,. . .,n). Let Δi be
the determinant obtained when substituting all elements in the ith column of
the fitness-matrix w(rij) (i,j = 1,. . .,n) with 1. The necessary and sufficient conditions for the existence of a stable equilibrium with all individual population
frequencies xi of the alleles being nonzero are then that T is negative definite
and that (−1)n − 1Δi > 0 for all i = 1,. . .,n. Geometrically, these first two conditions specify a peak in n-dimensional fitness space, and only one such peak is
allowed for all alleles to coexist (47).
For heterozygote advantage to be consequential (i.e., to be capable of
effectively stabilizing a balanced polymorphism against the stochastic fluctuations arising from random genetic drift), the fitness advantages of a heterozygote over its two homozygote have to be at least of order 1/N (10).
Because we are only interested in such consequential cases of heterozygote
advantage, we thus require, as a third condition, that for at least one pair of
alleles in a balanced polymorphism, it holds that w(rij) > max[w(rii),w(rjj)] + 1/N.
Sellis et al.
ACKNOWLEDGMENTS. We thank members of the D.A.P. laboratory, Daniel
Weinreich, Adam Eyre-Walker, Hunter Fraser, David Kingsley, Molly Przeworski, Matthew Stephens, Warren Ewens, Nick Barton, Kevin Bullaughey,
Marc Feldman, Daniel Fisher, Daniel Hartl, Ward Watt, and six anonymous
reviewers for helpful feedback. This research was supported by grants from
the National Institutes of Health Grant GM 077368 and National Science
Foundation Grant 0317171, CNS-0619926 (to D.A.P.). D.S. is a Stanford
Graduate Fellow. P.W.M. was a Human Frontier Science Program postdoctoral fellow.
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PNAS | December 20, 2011 | vol. 108 | no. 51 | 20671
EVOLUTION
In our simulations, we evaluate these three conditions for the fitness
matrix of all alleles with frequencies 0.05 < xi < 0.95. Negative definiteness of
T is tested by numerically calculating eigenvalues using symmetrical bidiagonalization with the QR reduction method (48) and checking for the
negativity of all eigenvalues. Signs of determinants Δi are estimated using
numerical LU decompositions (48).
All source code is openly available online at: http://sourceforge.net/projects/
fgm. Simulations were run on the Bio-X2 cluster at Stanford University.